Geometry & Dynamics Seminar 2013-14

The usual place and time are Tel Aviv University's Schreiber Building room 209, on Wednesdays at 14:10.

Please check each announcement since this is sometimes changed.

16.10.2013, 14:10 (Wednesday)	Orientation meeting for students
Location:	Schreiber bldg., room 209, Tel-Aviv University
20.10.2013, 11:10 (Sunday)	Lenya Ryzhik (Stanford University)
Title:	A wave and a particle in random media with slowly decaying correlations
Location:	Schreiber bldg., room 210, Tel-Aviv University
Abstract:	Most of the classical results on the evolution of waves and particles in random media focus on the regime when the random medium has rapidly decaying correlations, the canonical example being a random potential on a lattice with i.i.d. values at the vertices. A typical result in such setting is a central limit theorem for some physical observable (particle position, momentum etc.) arising at a universal time scale $T=\eps^{-2}$ where $\eps$ is the (small) amplitude of the fluctuation of the medium parameters. It turns out that when the medium fluctuations are far from i.i.d and have correlations that decay only algebraically in time and space, the picture is much richer - each observable picks its own time scale to get randomized  - the unity is lost and the orchestra plays out of sync. Time permitting,  I will illustrate this on the example of particles in random velocity fields, random Hamiltonians and  the Schroedinger equation. These results were obtained jointly with G.Bal, T. Komorowski, A. Novikov and S. Olla in various combinations.
23.10.2013, 14:10 (Wednesday)	Shiri Artstein-Avidan (Tel-Aviv University)
Title:	On symplectic isoperimetry, billiards, and Mahler's conjecture
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	We shall show how an isoperimetric conjecture regarding capacities of convex sets (called Viterbo's conjecture) implies the well known Mahler conjecture from convexity, regarding the minimizers, among symmetric convex bodies, of the volume product of a body and its polar. The connection uses some new results on billiard dynamics, which we shall explain as well. Based of joint work with Yaron Ostrover and with Roman Karasev.
28.10.2013, 16:10 (Monday)	Anton Izosimov (Moscow State University)
Title:	Algebraic geometry and stability for integrable systems
Location:	Schreiber bldg., room 210, Tel-Aviv University
Abstract:	In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of Theta functions of Riemann surfaces. However, the explicit formulas obtained in this way are of little or no use for solving such natural topological problems as the problem of Lyapunov stability. The goal of the talk is to demonstrate that these kind of problems can also be approached by means of classical algebraic geometry, and that this approach is very natural and fruitful. In particular, the stability problem for relative equilibria of the  free multidimensional rigid body will be considered.
30.10.2013, 15:10 (Wednesday)	Shimon Brooks (Bar Ilan University)
Title:	Quasimodes that do not Equidistribute
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The QUE Conjecture of Rudnick-Sarnak asserts that eigenfunctions of the Laplacian on Riemannian manifolds of negative curvature should equidistribute in the large eigenvalue limit.  For a number of reasons, it is expected that this property may be related to the (conjectured) small multiplicities in the spectrum.   One way to study this relationship is to ask about equidistribution for "quasimodes"--- or approximate eigenfunctions--- in place of highly-degenerate eigenspaces.   We will discuss the case of surfaces of constant negative curvature; in particular, we will explain how to construct examples of sufficiently weak quasimodes that do not satisfy QUE, and show how they fit into the larger theory.
6.11.2013, 14:10 (Wednesday)	Stepan Orevkov (Steklov Institute of Mathematics)
Title:	Algorithmic recognition of quasipositive braids of algebraic length two
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	A braid is called quasipositive if it is a product of standard generators of the braid group. We give an algorithm which decide if a given braid is a product of two conjugates of standard generators. The algorithm is based on the Garside theory which was developped for a solution of the conjugayc problem in the braid groups. We also give in the talk a short introduction to this theory.
20.11.2013, 14:10 (Wednesday)	Yochay Jerby (Tel Aviv University)
Title:	Exceptional collections and monodromies of Landau-Ginzburg equations.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Consider the collection of line bundles O,O(1),...,O(n) on n-dimensional complex projective space and let A be the endomorphism ring of its direct sum. A seminal result of Beilinson states that the bounded derived category of coherent sheaves on P^n is equivalent to the bounded derived category of right modules over A. Projective space is the "simplest" example of a toric manifold (it is the only toric manifold whose Picard group Pic(X) is of rank=1). A fundamental question asks which toric manifolds admit "exceptional" collections of line bundles, generalizing Beilinson's example? This question proves to be to algebraically hard, and is currently open. Recently, examples of toric manifolds which do not admit (full strongly) exceptional collections of line bundles were found by Hille & Perling and by Efimov, disproving previous conjectures on the subject. In this talk we would take a "mirror approach" to the question. We consider toric Fano manifolds. For such manifolds the theory of quantum cohomology associates a system of algebraic equations, known as the Landau-Ginzburg system of X (first introduced by V. Batyrev). We show that in low dimensions (two and three) t here exist striking relations between the solution set of the LG-system (denoted Crit(X)) and known examples of exceptional collections in Pic(X). For the considered examples we shall (a) show the existence of a map L: Crit(X) ---->Pic(X) whose image is an exceptional collection (b) show that, under the map L, monodromies of the Landau-Ginzburg system are related to quiver representations of the corresponding collection. These relations lead to postulate on the general, higher dimensional, case.
27.11.2013, 14:10 (Wednesday)	Lev Buhovski (Tel Aviv University)
Title:	Variations on Gromov-Eliashberg theorem
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	I will speak about flexibility and rigidity of symplectic homeomorphisms on smooth submanifolds. This subject arose from the celebrated Gromov-Eliashberg theorem, from the work by Opshtein on C^0 rigidity of characteristic foliation on smooth hypersurfaces, and from the work of Humiliere, Leclercq and Seyfaddini on C^0 coisotropic rigidity. Based on a joint work with Opshtein.
4.12.2013, 14:10 (Wednesday)	Oren Ben-Bassat
Title:	Shifted Symplectic Geometry
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	I will give a leisurely introduction to the shifted symplectic geometry of Pantev, Toen, Vaquie and Vezzosi. This material can be found at: http://arxiv.org/abs/1111.3209 I will emphasize shifted symplectic structures on moduli of bundles on Calabi-Yau spaces and will also discuss how to produce Lagrangians in these spaces. I will briefly discuss a new example from http://arxiv.org/abs/1309.0596 I will also discuss my joint work with Brav, Bussi and Joyce on canonical perverse sheaves on derived moduli stacks of vector bundles. I will not assume much knowledge of algebraic geometry, stacks, and derived geometry.
11.12.2013, 14:10 (Wednesday)	NO SEMINAR THIS WEEK!
18.12.2013, 14:10 (Wednesday)	Yael Karshon (University of Toronto)
Title:	Distinguishing symplectic blowups of the complex projective plane
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	A symplectic manifold that is obtained from CP^2 by k blowups is encoded by k+1 parameters: the size of the initial CP^2 and the sizes of the blowups.  We determine which values of these parameters yield symplectomorphic manifolds. This is joint work with Liat Kessler and Martin Pinsonnault.
25.12.2013, 14:10 (Wednesday)	Albert Fathi (École Normale Supérieure de Lyon)
Title:	Lyapunov forms
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Pursuing a work that was done with Pierre Pageault on Lyapunov functions for a dynamical system, we study the problem of existence of closed 1-forms for example for flows whose integral along orbits is non-positive, thus generalizing some work of Schwartzman--compare also with the work of Farber, Kappeler, Latschev,  and Zehnder. We will in fact consider not necessarily smooth closed 1-forms, but will also consider "forms" whose local primitives are merely continuous or Lipschitz functions.
31.12.2013, 11:10 (Tuesday)	Regina Rotman (University of Toronto)
Title:	Geometry of 2-dimensional spheres
Location:	Schreiber bldg., room 210, Tel-Aviv University
Abstract:	I will discuss some geometric inequalities that are valid for Riemannian manifolds diffeomorphic to the sphere of dimension 2. For example, consider the following basic question: Suppose a simple closed curve $\gamma$ on a Riemannian sphere M of diameter D can be contracted to a point in M over simple closed curves of length at most L. Is there a homotopy over loops based at some point of $\gamma$ that are not too long compared to L and D? The answer to this question is positive. (Joint with G. Chambers.) I will also prove that for any positive k and any two points of M there exist at least k geodesics connecting them of length at most 22kD. (Joint with A. Nabutovsky.)
8.1.2014, 14:10 (Wednesday)	Luis Haug (ETH)
Title:	Quantum homology of real Lagrangians in toric manifolds
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Consider a Fano toric symplectic manifold X, and let R be its real Lagrangian (the fixed point set under complex conjugation). We will explain the computation of the Lagrangian quantum homology ring QH(R) of R, and how it is related to the quantum homology ring QH(X) of the ambient symplectic manifold.
14.1.2014, 11:10 (Tuesday)	Luis Haug (ETH)
Title:	The Lagrangian cobordism group of the 2-torus
Location:	Schreiber bldg., room 210, Tel-Aviv University
Abstract:	The derived Fukaya category DFuk(M) of a symplectic manifold M is a triangulated category whose objects are the Lagrangian submanifolds in M. Recent work of Biran--Cornea provides a way of understanding cone decomposition in DFuk(M) via Lagrangian cobordisms. One consequence of their work is the existence of a canonical homomorphism from a naturally defined Lagrangian cobordism group \Omega_Lag(M) to the Grothendieck group K(DFuk(M)) of DFuk(M). We will prove that this map is an isomorphism when M is the standard symplectic 2-torus, using homological mirror symmetry.
15.1.2014, 14:10 (Wednesday)	Jake Solomon (Hebrew University)
Title:	Mirror symmetry and continuous symmetries
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Mirror symmetry is a duality between complex geometry on one manifold and symplectic geometry on another. I will discuss the symplectic mirror of a holomorphic C^* action on a complex manifold. The structure that arises is formulated in the language of symplectic cohomology and Lagrangian Floer cohomology. One byproduct is a deformation of the classical intersection number. The talk will not assume familiarity with mirror symmetry. This is joint work with P. Seidel.
19.02.2014, 14:10 (Wednesday)	Dmitry Turaev (Imperial College, London)
Title:	Fermi acceleration in time-dependent Hamiltonian systems
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	We discuss a novel theory of the evolution of energy in billiards with slowly moving boundaries, based on the Anosov-Kasuga theory of adiabatic invariants. We propose a stochastic description for the energy evolution, and demonstrate that the ergodicity of the billiard impedes the energy transfer from the moving boundary to the particle inside the billiard. We describe recent examples of time-dependent billiards for which a controlled violation of ergodicity leads to an exponential energy growth, i.e. to the effective energy transfer which is much more effective than in the ergodic case. We argue that the effect is of a fairly general nature: the periodic violation of ergodicity must be the key mechanism behind the energy flow from slow to fast degrees of freedom in slow-fast Hamiltonian systems with chaotic behaviour in the fast variables.
26.02.2014, 14:10 (Wednesday)	Misha Bialy (Tel Aviv University)
Title:	Invisibility for smooth Riemannian metrics
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	In this talk we give a construction of a smooth Riemannian metric on $\mathbf{R}^n$ which is standard Euclidean outside a compact set $K$ and such that it has $N={n(n+1)}/{2}$ invisible directions, meaning that all geodesics lines passing through the set $K$ in these directions remain the same straight lines on exit. For example in the plane our construction gives three invisible directions. This is in contrast with billiard type obstacles where a very sophisticated example due to A.Plakhov and V.Roshchina gives $2$ invisible directions in the plane and $3$ in the space.
04.03.2014, 11:10 (Tuesday)	Lior Bary-Soroker (Tel Aviv University)
Title:	Morse polynomials and Galois theory
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	In this talk I plan to describe the connection between univariate Morse polynomials and Galois theory. I will focus on the proof of the following theorem: Thm: Let x |--> f(x) be a polynomial map from the Riemann sphere to itself of degree n=deg f. Assume that f(x) is Morse (in the  sense that the critical points are non-degenerate and the critical values are distinct), then the monodromy group is the full symmetric group. The proof involves some geometry and some finite group theory.
05.03.2014, 14:10 (Wednesday)	Marcel Guardia (Université Paris 7)
Title:	Nearly integrable systems with orbits accumulating to KAM tori
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit. This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom  possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.
12.03.2014, 14:10 (Wednesday)	Laurent Charles (Universite de Paris VI)
Title:	Berezin-Toeplitz operators
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	In semi-classical analysis, Berezin-Toeplitz (BT) operators are similar to differential operators except that the underlying phase space is a Kähler compact manifold instead of a cotangent space. In this talk, I will review the basic properties of BT operators and the relationship with Kostant-Souriau prequantization. I will also present a "quantization commutes with reduction" theorem for BT operator. My motivating example will be the quantization of action coordinates in moduli spaces of polygons.
18.03.2014, 11:10 (Tuesday)	Laurent Charles (Universite de Paris VI)
Title:	Lagrangian states in Kähler quantization.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Lagrangian states are quantum states which concentrates in a particular way on a Lagrangian manifold. In this talk I will present their main characteristics and give applications to the spectral analysis of Berezin-Toeplitz operators and to the asymptotics of 6j-symbols.
19.03.2014, 14:10 (Wednesday)	Alexander Plakhov (University of Aveiro, Portugal)
Title:	Problems of optimal aerodynamic resistance and the Monge-Kantorovich problem
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	A rigid body moves in a rarefied medium of resting particles and at the same time very slowly rotates (somersaults). Each particle of the medium is reflected elastically when hitting the body boundary (multiple reflections are possible). The resulting resistance force acting on the body is time-dependent; we consider thetime-averaged value of resistance. The problem is: given a convex body, find a roughening of its surface that minimizes or maximizes its resistance. (The problem includes mathematical definition of roughening.) This problem is solved using the methods of billiard theory and optimal mass transportation. Surprisingly, the minimum and maximum depend only on the dimension of the ambient Euclidean space and do not depend on the original body. In particular, the resistance of a 3-dimensional convex body can be decreased by (approx.) 3.05% at most and can be increased at most twice by roughening.
19.03.2014, 15:10 (Wednesday)	Mikhail Lyubich (Stony Brook University)
Title:	Dynamics of dissipative polynomial automorphisms of C^2
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Two-dimensional complex dynamics is a recent area of research that has some similarities with the classical one-dimensional theory but also exhibits essentially new phenomena that require new analytic and dynamical tools. We will discuss recent advances in this field for moderately dissipative polynomial automorphisms of C^2. They include, in particular, a nearly complete description of the dynamics on periodic Fatou components and exploration of the bifurcation locus in holomorphic families of maps in question. Based on the joint work with Han Peters and Romain Dujardin.
23.04.2014, 14:10 (Wednesday)	Barak Weiss (Tel Aviv University)
Title:	Introduction to dynamics on moduli spaces of translation surfaces I.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The moduli space of translation surfaces arises naturally in several mathematical contexts: the study of billiards on rational polygons, complex analysis, asymptotic geometry of the mapping class group, and more. There is a natural SL(2,R) action on this space and in recent years a lot of effort has been invested in studying the dynamical properties of this action, both for its intrinsic interest and in view of applications to the topics above. In this sequence of two talks I will begin by defining the space and the action, and stating the main results about the dynamics. Then I will survey some applications to billiards in polygons.
30.04.2014, 14:10 (Wednesday)	Barak Weiss (Tel Aviv University)
Title:	Introduction to dynamics on moduli spaces of translation surfaces II.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The moduli space of translation surfaces arises naturally in several mathematical contexts: the study of billiards on rational polygons, complex analysis, asymptotic geometry of the mapping class group, and more. There is a natural SL(2,R) action on this space and in recent years a lot of effort has been invested in studying the dynamical properties of this action, both for its intrinsic interest and in view of applications to the topics above. In this sequence of two talks I will begin by defining the space and the action, and stating the main results about the dynamics. Then I will survey some applications to billiards in polygons.
07.05.2014, 14:10 (Wednesday)	Andrey E. Mironov (Sobolev Institute of Mathematics)
Title:	Commuting higher rank ordinary differential operators
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	We survey the theory of commuting ordinary differential operators. Some advances in the case of higher rank operators are discussed. Sufficient conditions are found when rank two operators are formally self-adjoint. We also point out examples of formally self-adjoint operators with polynomial coefficients, corresponding to a spectral curve of arbitrary genus. These operators define commutative subalgebra of the first Weyl algebra.
13.05.2014, 14:10 (Tuesday) SPECIAL COLLOQUIUM. NOTE THE DATE, TIME, AND PLACE!	Alejandro Uribe (University of Michigan)
Title:	The semiclassical limit in Bargmann space
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Bargmann space is a Hilbert space of holomorphic functions on C^n which can be used to formulate the quantum mechanics of a particle in R^n. In this talk I will show how Bargmann space is particularly well-suited to semiclassical analysis, which is the study of the asymptotics of quantum objects as Planck's constant tends to zero. I will emphasize the role played by coherent states and Lagrangian submanifolds, and discuss how approximating the quantum propagator in Bargmann space leads to extensions of the semiclassical trace formula.
14.05.2014, 14:10 (Wednesday)	Alejandro Uribe (University of Michigan)
Title:	The exponential map of the complexification of the group of hamiltonian diffeomorphisms.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	Donaldson has defined an object (a "formal Lie group") that behaves as the complexification of the group of hamiltonian diffeomorphisms of a Kähler manifold. Although not a group, it has a natural notion of exponential map. In this talk I will discuss geometric and analytic constructions of this map, in the real-analytic case.  I will discuss examples and applications (both current and potential) to geometry, geometric quantization, and the Toda PDE.
21.05.2014, 14:10 (Wednesday)	Nicolai Reshetikhin (UC Berkeley) (Blumenthal Lecture in Geometry)
Title:	Ice and 6-vertex models in statistical mechanics: mathematical perspective.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The talk will start with a historical overview of how the model appeared in statistical mechanics. This will be followed by an overview of some related combinatorial problems. After this the focus will shift to the "exact solution" of the model with periodic boundary conditions followed by the discussion of algebraic structures involved in the exact solution (quantum groups with braiding). Then there will be a brief discussion of other boundary conditions and of limit shape phenomenon. If time permit, we will see how the ASEP (asymmetric Exclusion Process) fits into the picture.
28.05.2014, 14:10 (Wednesday)	Stephen Kleene (M.I.T)
Title:	Singular Perturbation for Mean Curvature Type Equations.
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	I will discuss my recent work with  Niels Martin Moller and Nicos Kapouleas in which we applied singular perturbation techniques to construct the first examples of complete embedded self shrinkers  with high topology. The techniques are quite general and somewhat ad hoc, and (time permitting ) I will discuss our current work in formalizing and understanding the limits of the process.
11.06.2014, 14:10 (Wednesday)	Francois Lalonde (Universite de Montreal)
Title:	On the genus g-area in symplectic topology
Location:	Schreiber bldg., room 209, Tel-Aviv University
Abstract:	The genus g-area measures a sort of geometric norm associated to each Hamiltonian diffeomophism of a symplectic manifold. It is obtained by attaching g handles on the disc that constitutes the base of a "cylinder" that measures the geometric Hofer norm -- when g = 0, this boils down to the geometric Hofer norm of Lalonde-McDuff. In this talk, we introduce the relevant definitions and prove the main theorem concerning this norm, that is to say that it measures exactly the distance, in Hofer's norm, between a given Hamiltonian diffeomorphism and the space of all Hamiltonian diffeomorphisms of commutator length smaller or equal to $g$. While some of the conjectures on this question were formulated by the speaker in 2004, and half of them were proved with Andrei Teleman in 2012, the last and most important step was settled in joint work with Egor Shelukhin whose contribution was fundamental.